| L(s) = 1 | + (0.980 − 4.92i)3-s + (3.33 − 2.22i)5-s + (9.23 + 6.17i)7-s + (−15.0 − 6.22i)9-s + (2.99 − 0.596i)11-s + (−17.0 − 17.0i)13-s + (−7.70 − 18.6i)15-s + (−3.78 + 16.5i)17-s + (−6.94 + 2.87i)19-s + (39.4 − 39.4i)21-s + (3.61 + 18.1i)23-s + (−3.42 + 8.25i)25-s + (−20.2 + 30.3i)27-s + (12.1 + 18.1i)29-s + (33.7 + 6.71i)31-s + ⋯ |
| L(s) = 1 | + (0.326 − 1.64i)3-s + (0.666 − 0.445i)5-s + (1.31 + 0.881i)7-s + (−1.66 − 0.691i)9-s + (0.272 − 0.0541i)11-s + (−1.31 − 1.31i)13-s + (−0.513 − 1.24i)15-s + (−0.222 + 0.974i)17-s + (−0.365 + 0.151i)19-s + (1.87 − 1.87i)21-s + (0.156 + 0.789i)23-s + (−0.136 + 0.330i)25-s + (−0.751 + 1.12i)27-s + (0.418 + 0.626i)29-s + (1.08 + 0.216i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0284 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0284 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.29001 - 1.25383i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.29001 - 1.25383i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + (3.78 - 16.5i)T \) |
| good | 3 | \( 1 + (-0.980 + 4.92i)T + (-8.31 - 3.44i)T^{2} \) |
| 5 | \( 1 + (-3.33 + 2.22i)T + (9.56 - 23.0i)T^{2} \) |
| 7 | \( 1 + (-9.23 - 6.17i)T + (18.7 + 45.2i)T^{2} \) |
| 11 | \( 1 + (-2.99 + 0.596i)T + (111. - 46.3i)T^{2} \) |
| 13 | \( 1 + (17.0 + 17.0i)T + 169iT^{2} \) |
| 19 | \( 1 + (6.94 - 2.87i)T + (255. - 255. i)T^{2} \) |
| 23 | \( 1 + (-3.61 - 18.1i)T + (-488. + 202. i)T^{2} \) |
| 29 | \( 1 + (-12.1 - 18.1i)T + (-321. + 776. i)T^{2} \) |
| 31 | \( 1 + (-33.7 - 6.71i)T + (887. + 367. i)T^{2} \) |
| 37 | \( 1 + (-11.5 + 58.0i)T + (-1.26e3 - 523. i)T^{2} \) |
| 41 | \( 1 + (-16.6 - 11.1i)T + (643. + 1.55e3i)T^{2} \) |
| 43 | \( 1 + (-27.2 - 11.2i)T + (1.30e3 + 1.30e3i)T^{2} \) |
| 47 | \( 1 + (14.3 + 14.3i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (65.0 - 26.9i)T + (1.98e3 - 1.98e3i)T^{2} \) |
| 59 | \( 1 + (3.38 - 8.17i)T + (-2.46e3 - 2.46e3i)T^{2} \) |
| 61 | \( 1 + (46.1 - 69.0i)T + (-1.42e3 - 3.43e3i)T^{2} \) |
| 67 | \( 1 + 46.3iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (-11.2 + 56.7i)T + (-4.65e3 - 1.92e3i)T^{2} \) |
| 73 | \( 1 + (-90.6 + 60.5i)T + (2.03e3 - 4.92e3i)T^{2} \) |
| 79 | \( 1 + (-18.6 + 3.70i)T + (5.76e3 - 2.38e3i)T^{2} \) |
| 83 | \( 1 + (-51.2 - 123. i)T + (-4.87e3 + 4.87e3i)T^{2} \) |
| 89 | \( 1 + (36.7 - 36.7i)T - 7.92e3iT^{2} \) |
| 97 | \( 1 + (24.7 + 37.1i)T + (-3.60e3 + 8.69e3i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.56485022469119386176985993930, −12.22545192550801825424236294216, −10.90575703314422749104136128362, −9.273783255090535077438048227918, −8.220768692439190036766119991508, −7.60248635478782493948333549534, −6.09862051008090518541897743673, −5.15621087875610906565540076550, −2.50020348043845063726629712592, −1.43698350108213542912709186672,
2.48105495323104770576327638261, 4.41443906445118112904879236647, 4.77862951607104037684438621974, 6.69495353157090294355514977221, 8.138132872796737053599882286389, 9.417606411859719839083767942906, 10.03489302044856349493449388424, 10.97430670004333267570411350164, 11.79503919163775702547043144658, 13.89509571957391534525459198289
